2.2.5.5. Static Friction

Table of Contents

What static friction is

Static friction is the frictional force that acts between two surfaces that are in contact and are not sliding relative to each other. Its role is to prevent slipping when some other force tries to make one surface move across the other.

A very important point is that static friction does not always have one fixed value. It adjusts itself to whatever value is needed to keep the surfaces from sliding, up to a certain maximum.

If you gently push a heavy box and it does not move, static friction is present. If you push harder and the box still does not move, static friction becomes larger. Only when your push becomes too large does the box begin to slide.

Static friction opposes the tendency of relative motion between surfaces, not necessarily the overall motion of the object.

How static friction behaves

Suppose a block rests on a horizontal floor. If you apply a small horizontal force, the block may remain at rest. In that case, static friction acts in the opposite direction and matches your applied force in magnitude.

If the applied force is $F$, and the block does not move, then the static friction force $f_s$ satisfies

$$
f_s = F
$$

as long as this required friction is not too large.

There is, however, a maximum possible static friction force. It is given by

$$
f_{s,\max} = \mu_s N
$$

where $N$ is the normal force and $\mu_s$ is the coefficient of static friction.

So the full rule is

$$
0 \le f_s \le \mu_s N
$$

This means static friction can take any value from zero up to its maximum value.

The correct inequality for static friction is
$$
f_s \le \mu_s N
$$
It is not always true that
$$
f_s = \mu_s N
$$
Equality holds only when slipping is about to begin.

The coefficient of static friction

The coefficient of static friction, written $\mu_s$, depends on the pair of surfaces in contact. Rough surfaces usually have a larger value than smooth surfaces, but the exact value depends on the materials and surface conditions.

It has no unit, because it is a ratio.

A larger $\mu_s$ means the surfaces can resist a greater tangential force before sliding starts.

Static friction on a horizontal surface

For a block on a level floor, the normal force is often equal to the weight if there are no other vertical forces:

$$
N = mg
$$

Then the maximum static friction becomes

$$
f_{s,\max} = \mu_s mg
$$

If the applied horizontal force is less than this value, the block remains at rest. If it becomes greater, static friction can no longer hold the block, and sliding begins.

Rendering scientific sketch…

Block on a horizontal surface with static friction

Static friction on an inclined surface

Static friction is also what keeps an object from sliding down a slope. On an incline, the component of the weight parallel to the surface tends to pull the object downhill. Static friction acts uphill to oppose that tendency.

If the incline angle is $\theta$, then the downhill component of weight is

$$
mg \sin\theta
$$

and the normal force is

$$
N = mg \cos\theta
$$

So the maximum static friction is

$$
f_{s,\max} = \mu_s mg \cos\theta
$$

The object remains at rest as long as

$$
mg \sin\theta \le \mu_s mg \cos\theta
$$

or equivalently,

$$
\tan\theta \le \mu_s
$$

This explains why an object can stay at rest on a gentle incline but starts to slip on a steeper one.

Rendering scientific sketch…

Static friction on an incline

Direction of static friction

The direction of static friction must always be worked out from the situation. It acts in the direction that prevents relative slipping.

This can sometimes be surprising. For example, if a wheel rolls without slipping, static friction may point forward or backward depending on whether the wheel is being driven or simply coasting. The direction is determined by the tendency of the contact point to slip.

Do not guess the direction of static friction from motion alone. Determine which way the surfaces would slip if there were no friction, then static friction acts opposite to that tendency.

Static friction compared with kinetic friction

Static friction acts when there is no slipping. Kinetic friction acts when surfaces are sliding past each other. Usually,

$$
\mu_s > \mu_k
$$

so it is often harder to start motion than to keep motion going.

This is why a box may require a strong initial push to begin moving, but once it slides, a smaller force can keep it moving.

Type of friction	Condition	Typical size
Static friction	No slipping	$0 \le f_s \le \mu_s N$
Kinetic friction	Sliding occurs	$f_k = \mu_k N$

How to solve problems with static friction

When analyzing a problem, first assume the object does not slip. Then use Newton's laws to find the friction force required for equilibrium or for the given motion. After that, check whether this required value is less than or equal to $\mu_s N$.

If it is, the assumption of no slipping is valid. If it is not, static friction is not enough, and the object must slide.

This method is very useful because static friction is not known in advance. It must be found from the conditions of the problem, then checked against the maximum allowed value.

Problem solving rule for static friction:

Assume no slipping.
Compute the needed friction force.
Check whether
$$
f_s \le \mu_s N
$$
If not, slipping occurs.

Everyday examples

Walking is possible because of static friction between your shoes and the ground. Your foot pushes backward on the ground, and static friction from the ground pushes you forward.

A parked car on a hill remains in place because static friction prevents the tires from slipping.

A ladder leaning against a wall may stay at rest because static friction at the floor, or at both contact surfaces, prevents sliding.

Key ideas to remember

Static friction is a responsive force. It does whatever is necessary to prevent slipping, but only up to a limit set by $\mu_s N$.

The most important feature is that its actual value is usually not known beforehand.

Essential facts:
$$
0 \le f_s \le \mu_s N
$$
Maximum static friction:
$$
f_{s,\max} = \mu_s N
$$
Static friction acts to prevent relative slipping between surfaces.

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